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Astérisque - Parutions - 393 (2017)

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The realization space of an unstable coalgebra
Georg Biedermann, Georgios Raptis, Manfred Stelzer
Astérisque 393 (2017), viii+148 pages
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Résumé :
Espaces de réalisation d'une coalgèbre instable
L'homologie singulière à coefficients dans un corps premier d'un espace topologique a la structure d'une coalgèbre instable sur l'algèbre de Steenrod. La question de savoir si une coalgèbre instable donnée est isomorphe à l'homologie d'un espace topologique est le problème de réalisation. Nous décrivons une tour d'espaces qui converge vers l'espace de module des réalisations sous des conditions raisonables. La différence entre deux niveaux consécutifs est controllée par la cohomologie des coalgèbres instables. Ces résultats sont obtenus par une étude comparative approfondie des théories d'homotopie des coalgèbre instables cosimpliciales et des espaces cosimpliciaux.

Mots-clefs : Espace de module, homologie singulière, algèbre de Steenrod, coalgèbre instable, cohomologie d'André-Quillen, théorie des obstructions, spiral exact sequence.

Abstract:
Unstable coalgebras over the Steenrod algebra form a natural target category for singular homology with prime field coefficients. The realization problem asks whether an unstable coalgebra is isomorphic to the homology of a topological space. We study the moduli space of such realizations and give a description of this in terms of cohomological invariants of the unstable coalgebra. This is accomplished by a thorough comparative study of the homotopy theories of cosimplicial unstable coalgebras and of cosimplicial spaces.

Keywords: Moduli space, singular homology, Steenrod algebra, unstable coalgebra, André-Quillen cohomology, obstruction theory, spiral exact sequence.

Class. math. : 55S10, 55S35, 55N10, 55P62


ISBN : 978-2-85629-868-8
ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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